In this paper, matrices are square unless stated otherwise. We use \(I_{n}\)  and \(J_{n}\)  to denote the \(n \times n\)  identity and all-ones matrices respectively, and drop the subscript when the order is clear from context. Recall that a matrix is Hermitian if \(G^{\ast } = G\) , and positive definite if its eigenvalues are positive real numbers.
The Gram matrix of \(M\)  is the matrix \(MM^{\ast }\) , which has as entries the inner products of rows of \(M\) .
A positive definite Hermitian matrix \(G\)  is a Gram matrix: via the square root of a positive matrix, it can be shown that
there exists a matrix \(X\)  such that \(XX^{\ast } = G\) . Conversely, the Gram matrix of a set of linearly independent vectors is
Hermitian positive definite. There is a well-developed theory for positive definite matrices;
see for example the monograph of Horn and Johnson [1]}. We follow Hadamard in considering a
minor of order \(k\)  to be the determinant of a \(k\times k\)  submatrix.
